PS3 Solutions
1.
11.2B from the book: Consider again (as in the previous problem) the production function x = f
(
ℓ
) = 100
ℓα
.
a.
Derive the firm’s cost function.
Answer: First, we need to invert the production function. We thus take x = 100
ℓα
and solve
for
ℓ
to get
Then we multiply this by w to get
b.
Derive the marginal and average cost functions and determine how their relationship
to one another differs depending on
α
.
Answer: The marginal cost function is:
The average cost function is:
Notice that AC = MC only when x = 0 unless
α
= 1 in which case MC = AC everywhere.
This is because when
α
= 1, the production process is linear
—
which means the
marginal cost is constant. When
α
< 1, MC > AC everywhere other than when x = 0; and
when
α
> 1, MC < AC everywhere other than when x = 0. This is because the former
represents cases where marginal product of labor diminishes throughout while the
latter represents cases where the marginal product of labor increases throughout.
c.
What is the supply function for this firm when
α
= 0.5
? What is the firm’s labor
demand curve?
Answer: In this case MC > AC everywhere
—
so the MC curve is in fact the supply curve.
Put differently, we can set p equal to MC and solve for the output supply of
When
α
= 0.5, this reduces to
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Substituting this into the labor function above and setting
α
= 0.5 in that equation, we
then get the labor demand as
d.
How do your answers change when
α
= 1.5?
Answer: When
α
> 1, MC < AC throughout
—
which implies there is no supply curve. The
reason is that in this case we have increasing marginal product of labor throughout
—
which
implies the pricetaking firm would wish to produce an infinite quantity of the output.
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 Spring '08
 Woroch
 Economics, Microeconomics, Supply And Demand, Economics of production

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